left|begin{array}{ccc} 1 & 1 & 1 a^2 & b^2 & | vedclass

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(C) Let $\Delta = \begin{vmatrix} 1 & 1 & 1 \ a^2 & b^2 & c^2 \ a^3 & b^3 & c^3 \end{vmatrix}$.Apply $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_1$:$\D

Vedclass · Accepted Answer

$(ab+bc+ca)(a-b)(b-c)(c-a)$

Vedclass · Answer

(C) Let $\Delta = \begin{vmatrix} 1 & 1 & 1 \ a^2 & b^2 & c^2 \ a^3 & b^3 & c^3 \end{vmatrix}$.Apply $C_2 	o C_2 - C_1$ and $C_3 	o C_3 - C_1$:$\Delta = \begin{vmatrix} 1 & 0 & 0 \ a^2 & b^2 - a^2 & c^2 - a^2 \ a^3 & b^3 - a^3 & c^3 - a^3 \end{vmatrix}$$= \begin{vmatrix} (b-a)(b+a) & (c-a)(c+a) \ (b-a)(b^2+ab+a^2) & (c-a)(c^2+ac+a^2) \end{vmatrix}$$= (b-a)(c-a) \begin{vmatrix} b+a & c+a \ b^2+ab+a^2 & c^2+ac+a^2 \end{vmatrix}$$= (b-a)(c-a) [(b+a)(c^2+ac+a^2) - (c+a)(b^2+ab+a^2)]$$= (b-a)(c-a) [bc^2+abc+a^2b+ac^2+a^2c+a^3 - (cb^2+abc+a^2c+ab^2+a^2b+a^3)]$$= (b-a)(c-a) [bc^2+ac^2-cb^2-ab^2]$$= (b-a)(c-a) [bc(c-b) + a(c^2-b^2)]$$= (b-a)(c-a) [bc(c-b) + a(c-b)(c+b)]$$= (b-a)(c-a)(c-b) [bc + ac + ab]$$= (a-b)(b-c)(c-a)(ab+bc+ca)$.

$\left|\begin{array}{ccc} 1 & 1 & 1 \\ a^2 & b^2 & c^2 \\ a^3 & b^3 & c^3 \end{array}\right|=$

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